Parabolic Parabolic Partial Differential Equations Research Articles

Numerical simulation and applications of the convection–diffusion–reaction equation with the radial basis function in a finite-difference mode

This paper develops two local mesh-free methods for designing stencil weights and spatial discretization, respectively, for parabolic partial differential equations (PDEs) of convection–diffusion–reaction type. These are known as the radial-basis-function generated finite-difference method and the Hermite finite-difference method. The convergence and stability of these schemes are investigated numerically using some examples in two and three dimensions with regularly and irregularly shaped domains. Then we consider the numerical pricing of European and American options under the Heston stochastic volatility model. The European option leads to the solution of a two-dimensional parabolic PDE, and the price of the American option is given by a linear complementarity problem with a two-dimensional parabolic PDE of convection–diffusion–reaction type. Then we use the operator splitting method to perform time-stepping after space discretization. The resulting linear systems of equations are well conditioned and sparse, and by numerical experiments we show that our numerical technique is fast and stable with respect to the change in the shape parameter of the radial basis function. Finally, numerical results are provided to illustrate the quality of approximation and to show how well our approach converges with the results presented in the literature.

Direct and inverse source problems for degenerate parabolic equations

Abstract Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications.

Input-to-State Stability for a Class of One-Dimensional Nonlinear Parabolic PDEs with Nonlinear Boundary Conditions

The aim of this paper is to introduce a weak maximum principle--based approach for studying the input-to-state stability (ISS) with respect to boundary disturbances and states in certain classes for a class of one-dimensional nonlinear parabolic partial differential equations (PDEs) with nonlinear boundary conditions. To tackle the difficulties in ISS analysis due to, in particular, the nonlinear terms on the boundary, we establish first several maximum estimates for the solutions of linear parabolic PDEs with different nonlinear boundary conditions by means of the weak maximum principle. Then, using the technique of splitting and combining maximum estimates for the solutions of linear parabolic PDEs and the Lyapunov method, we establish ISS estimates for nonlinear parabolic PDEs with nonlinear boundary conditions. Two examples of specific parabolic equations with nonlinear boundary conditions are provided to illustrate the developed approach.

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems. In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial derivatives in the financial industry. The PDEs appearing in financial engineering applications are often nonlinear and high dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue is that most approximation methods for nonlinear PDEs in the literature suffer under the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods have not been shown not to suffer under the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend those findings to a more general class of semilinear PDEs including as special cases semilinear Black-Scholes equations used for the pricing of financial derivatives with default risks. More specifically, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove that the computational effort of our method grows at most polynomially both in the dimension and the reciprocal of the prescribed approximation accuracy. This is, to the best of our knowledge, the first result showing that the approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.

Observer Design for Lipschitz Nonlinear Parabolic PDE Systems With Unknown Input

In this article, a novel method to design the observer for a class of uncertain Lipschitz nonlinear parabolic partial differential equations (PDE) systems is investigated. First, the observer and the dynamic errors with undetermined parameters for the parabolic PDE systems subject to appropriate boundary conditions are presented. The conditions of the designed observer are involved. Then the analysis of asymptotic stability and $\mathcal {H}_\infty $ performance conditions for the observer design of uncertain nonlinear parabolic PDE systems are studied and presented in terms of matrix inequalities based on the Lyapunov stability theory. Finally, the effectiveness of the proposed method is validated by a numerical parabolic PDE system.

Switching state observer design for semilinear parabolic PDE systems with mobile sensors

This paper studies the problem of state observation for a class of semilinear parabolic partial differential equation (PDE) systems using mobile sensors, where the spatial domain is decomposed into multiple subdomains according to the number of sensors and each sensor is able to move within the respective spatial subdomain. Initially, the well-posedness of the original parabolic PDE system is discussed. Subsequently, a mode indicator function is used to indicate the different modes for all sensors and a mode-dependent observation scheme is presented to reduce the conservatism of observer design. Then, by employing Lyapunov direct technique and Wirtinger’s inequality, an integrated design of switching state observers and mobile sensor guidance laws is developed in terms of linear matrix inequalities, such that the resulting state estimation error system is exponentially stable. Moreover, the mobile sensor guidance can enhance the transient performance of the error system. Finally, numerical simulations are given to illustrate the effectiveness of the proposed design method.

A unified Lyapunov-based design for a dynamic compensator of linear parabolic MIMO PDEs

This paper addresses the problem of dynamic compensator design for exponential stabilisation of linear parabolic partial differential equations (PDEs) with multiple actuation control inputs and multiple non-collocated observation outputs. Both in-domain control and boundary control are considered. A new observer-based dynamic compensator is constructed by the non-collocated observation outputs such that the resulting closed-loop coupled PDEs are exponentially stable. By constructing a Lyapunov function candidate and using Poincaré–Wirtinger inequality's variants, a sufficient condition for the existence of such dynamic compensator is presented in terms of standard linear matrix inequalities (LMIs). The closed-loop well-posedness analysis result is also established by the method of -semigroup and its perturbations by bounded/unbounded linear operators. Finally, numerical simulation results are presented to support the proposed design method.

Sampled-Data Fuzzy Control With Guaranteed Cost for Nonlinear Parabolic PDE Systems via Static Output Feedback

This article introduces a sampled-data (SD) static output feedback fuzzy control (FC) with guaranteed cost for nonlinear parabolic partial differential equation (PDE) systems. First, a Takagi–Sugeno (T–S) fuzzy parabolic PDE model is employed to represent the nonlinear PDE system. Second, with the aid of the T–S fuzzy PDE model, a SD FC design with guaranteed cost under spatially averaged measurements is developed in the formulation of linear matrix inequalities by utilizing a time-dependent Lyapunov functional and inequality techniques, which can stabilize exponentially the PDE system while providing an optimized upper bound on the cost function. The membership functions of the proposed controller are determined by the measurement output and independent of the fuzzy PDE plant model. Finally, simulation results are presented to control the diffusion equation and the FitzHugh–Nagumo equation for demonstrating the effectiveness of the proposed method.

Sampled-data fuzzy control with space-varying gains for nonlinear time-delay parabolic PDE systems

This paper introduces a sampled-data fuzzy control (SDFC) with space-varying gains for nonlinear time-delay parabolic partial differential equation (PDE) systems. First, a Takagi-Sugeno (T-S) fuzzy time-delay parabolic PDE model is employed to represent the nonlinear time-delay PDE system. Second, with the aid of the T-S fuzzy time-delay PDE model, a SDFC design with space-varying gains is developed in the formulation of space-dependent linear matrix inequalities (LMIs) by constructing an appropriate Lyapunov functional, which can stabilize exponentially the time-delay PDE system. These stabilization conditions can be applied to either slowing-varying time delay or fast-varying one. Finally, simulation results on two examples are presented to demonstrate the effectiveness of the proposed method.

Mobile observation for distributed parameter system with moving boundary over mobile sensor networks

This paper considers the state observation for a class of distributed parameter systems (DPSs) with moving boundaries modelled by parabolic partial differential equations (PDEs). The method of mobile observation is presented to improve the observation performance of the systems and to eliminate the influence of moving boundaries on system observation with the aids of mobile sensor networks (MSNs). The MSNs which can move throughout the time-dependent spatial domain are used to provide the spatially averaged observations of the DPSs. By using the abstract evolution equation theory of parabolic PDEs and the Lyapunov stability arguments, a centralised observer is designed and a mobile control scheme for each of the mobile sensors is presented while taking account of the dynamics of the MSNs. A numerical example is finally presented to illustrate the effectiveness and the advantages of the proposed approach.

Lyapunov-based boundary feedback design for parabolic PDEs

This paper presents a methodology for the construction of simple Control Lyapunov Functionals (CLFs) for boundary controlled parabolic Partial Differential Equations (PDEs). The proposed methodology provides functionals that contain only simple (and not double or triple) integrals of the state. Moreover, the constructed CLF is ‘almost diagonal’ in the sense that it contains only a finite number of cross-products of the (generalised) Fourier coefficients of the state. The methodology for the construction of a CLF is combined with a novel methodology for boundary feedback design in parabolic PDEs. The proposed feedback design methodology is Lyapunov-based and the feedback controller is an ‘integral’ controller with internal dynamics. It is also shown that the obtained simple CLFs can provide nonlinear boundary feedback laws which achieve global exponential stabilisation of semilinear parabolic PDEs with nonlinearities that satisfy a linear growth condition.

A class of numerical algorithms based on cubic trigonometric B-spline functions for numerical simulation of nonlinear parabolic problems

In this work, the authors developed two new B-spline collocation algorithms based on cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic partial differential equations (PDEs) with Dirichlet and Neumann boundary conditions. In the first algorithm, cubic trigonometric B-spline functions are directly used for approximate solutions of parabolic PDEs with Neumann boundary conditions. But, the Dirichlet boundary conditions cannot be handled directly by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and the second algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithms reduced the parabolic problem into a system of first-order nonlinear ordinary differential equations (ODEs) in time variable. Then, strong stability-preserving-Runge–Kutta3 (SSP-RK3) scheme is used to solve the obtained system. Some well-known parabolic problems are solved to check the accuracy and efficiency of the proposed algorithms. The algorithms can be extended to solve multidimensional problems arising as model equations in physical, chemical and biophysical phenomenon.

A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis

We consider a non-linear parabolic partial differential equation (PDE) on Rd with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition and blow-up times. We prove a global existence and uniqueness result assuming further properties on the non-linearity. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

Fuzzy Stabilization Design for Semilinear Parabolic PDE Systems With Mobile Actuators and Sensors

This paper deals with a fuzzy stabilization design problem for a class of semilinear parabolic partial differential equation (PDE) systems using mobile actuators and sensors. Initially, a Takagi–Sugeno (T–S) fuzzy PDE model is employed to accurately represent the semilinear parabolic PDE system. Subsequently, based on the T–S fuzzy model, a stabilization scheme containing the fuzzy controllers and the guidance of mobile actuator/sensor pairs is proposed, where the spatial domain is decomposed into multiple subdomains according to the number of actuator/sensor pairs and each actuator/sensor pair is capable of moving within the respective subdomain. Then, by a Lyapunov direct technique, an integrated design of fuzzy controllers plus mobile actuator/sensor guidance laws is developed in the form of bilinear matrix inequalities (BMIs), such that the resulting closed-loop system is exponentially stable and the mobile actuator/sensor guidance can enhance the transient performance of the closed-loop system. Furthermore, an iterative algorithm based on linear matrix inequalities is proposed to solve the BMIs. Finally, two examples are given to illustrate the effectiveness of the proposed method.

A Lyapunov-based design of dynamic feedback compensator for linear parabolic MIMO PDEs

Abstract This paper discusses dynamic feedback compensator design for a linear parabolic partial differential equation (PDE) with multiple inputs and multiple outputs. Actuating control inputs are provided by actuators distributed over partial areas (or active at specified positions) of the spatial domain, and observation outputs are taken from the non-collocated sensors distributed over partial areas of the spatial domain. An observer-based dynamic feedback compensator is constructed via the observer-based feedback control technique to exponentially stabilize the multi-input–multi-output PDE in the spatial $\mathscr{L}^2$ norm. By constructing an appropriate Lyapunov function candidate and using two variants of Poincaré–Wirtinger inequality, sufficient conditions on the existence of such observer-based dynamic feedback compensator are developed and presented in terms of linear matrix inequalities. The well posedness of the closed-loop coupled PDEs is also analyzed within the framework of $C_0$ semigroup theory. Finally, numerical simulation results are given to show the effectiveness of the proposed method.

Extended BACOLI

BACOLI is a Fortran software package for solving one-dimensional parabolic partial differential equations (PDEs) with separated boundary conditions by B-spline adaptive collocation methods. A distinguishing feature of BACOLI is its ability to estimate and control error and correspondingly adapt meshes in both space and time. Many models of scientific interest, however, can be formulated as multiscale parabolic PDE systems, that is, models that couple a system of parabolic PDEs describing dynamics on a global scale with a system of ordinary differential equations describing dynamics on a local scale. This article describes the Fortran software eBACOLI, the extension of BACOLI to solve such multiscale models. The performance of the extended software is demonstrated to be statistically equivalent to the original for purely parabolic PDE systems. Results from eBACOLI are given for various multiscale models from the extended problem class considered.

On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in the state-of-the-art pricing and hedging of financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article [E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)] we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution for semilinear heat equations that the computational complexity is bounded by $O( d \, \epsilon^{-(4+\delta)})$ for any $\delta\in(0,\infty)$, where $d$ is the dimensionality of the problem and $\epsilon\in(0,\infty)$ is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of $100$-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for these 100-dimensional example PDEs are very satisfactory in terms of accuracy and speed. In addition, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the literature.

Finite Element Methods for Partial Differential Equations for Option Pricing

This manuscript gives introduction to parabolic Partial Differential Equations (PDE) in one-dimensional case. It recalls the notion and main limitations of Black-Scholes model and establishes the Black-Scholes formula. It recalls the standard machinery of solving the parabolic PDE which is well-known in standard literature such as text-book of Bass (1999). Here we recall the notion of infinitesimal generator of the diffusion process and recall some basic well-known results on the infinitesimal generator of the Markov process. We avoid giving complete proofs of known results and provide reader with a short hint of proofs. Here, we recall again the theory of solving the parabolic PDE, we recall machinery of changes of variables and unknown functions. Then, we derive the Black-Scholes formula and give some intuition of application to some classes of options, such as a Vanilla European Call and Put. The next section is devoted to giving some notions of the classical solution to the parabolic PDE. It is immediately followed by the variational framework. The introduction to the reader of weighted Sobolev spaces on which the solution to the class of parabolic PDE under study lives is a subject to the next section in this Chapter. Straight after that we give a notion of weak formulation of the Black-Scholes equation followed by proving the regularity property of the weak solutions, application of the maximum principle for weak solutions which is a powerful tool to provide the solution to parabolic PDE and work further on derivation of various bounds. We give more insight into the localization procedure. We apply this methodology to option pricing, via Put-Call parity relation we derive a payoff function for a European Vanilla Put and Call. Among the further examples are General European Options, Barrier Options, European Options on a Basket of Asset, American Options, Asian Options. The chapter continues with giving some introduction to application of Finite Element methods where we recall the main steps of using this machinery. We recall as well more innovative Finite Element methods for the class of stochastic volatility models.

Consensus of uncertain parabolic PDE agents via adaptive unit‐vector control scheme

This study focuses on the asymptotical consensus and synchronisation for coupled uncertain parabolic partial differential equation (PDE) agents with Neumann boundary condition (or Dirichlet boundary condition) and subject to a distributed disturbance whose norm is bounded by a constant which is not known a priori. Based on adaptive distributed unit-vector control scheme and Lyapunov functional scheme, the distributed unit-vector controller with adaptive gain is suggested to provide for the leader–following tracking of uncertain parabolic PDE agents. The suggested gradient-based and mono-directional gain adaptation provides convergence without requiring the knowledge of an upper bound to the norm of the external disturbance. The simulation is proposed to account for the effectiveness and robustness of the provided method.

Observer‐based output feedback compensator design for linear parabolic PDEs with local piecewise control and pointwise observation in space

This study presents an observer-based output feedback compensator design for a linear parabolic partial differential equation (PDE), where a finite number of actuators and sensors are active over partial areas and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger-type PDE observer is first constructed by using the non-collocated pointwise observation to exponentially track the PDE state. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing a Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for definite integrals, sufficient conditions on the exponential stability of the resulting closed-loop coupled PDEs are presented in terms of standard linear matrix inequalities. Furthermore, both open-loop and closed-loop well-posedness analysis results are also established by the C 0 -semigroup method. Numerical simulation results are presented to show the effectiveness of the proposed design method.